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Prove that the graph of a measurable function is measurable

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Homework Statement


Let f: X->R be measurable, prove that Z={(x,y)|y=f(x)} is a measurable set of XxR.


Homework Equations



A subset Z of XxR is measurable iff Z is a countable union of product of measurable sets of X and R.

The Attempt at a Solution



Let [itex]R=\cup_kV_k[/itex], where [itex]V_k[/itex] are finite intervals of R, let [itex]U_k=f^{-1}(V_k)[/itex], obviously [itex]Z\subset\cup_kU_k\times V_k[/itex] where [itex]\cup_kU_k\times V_k[/itex] is measurable since both [itex]U_k[/itex] and [itex]V_k[/itex] are measurable. How to I proceed to show that Z as a subset of a measurable set of XxR is also measurable?
 

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